Summary: Math 2210 Problem Set 3
Key
Problem 1 Find a space curve r(t) that is a spiral of radius 2 centered on the x axis.
Parameterize your spiral so that the derivative of r projected onto i is constant.
Solution 1: r(t) = (t, 2Cos(t), 2Sin(t)) To see this is correct note the derivative in
direction i is 1 and that the y and z coordinates make a circle of radius 2.
Problem 2 Give the type (e.g. "circle") for each of the following space curves.
(i) r(t) = (t2
, 2t + 1, t - 5),
(ii) s(t) = (1, 2Sin(t), 5Cos(t)),
(iii) t(t) = (0, 5Sin(t), 5Cos(t)), and
(iv) u(t) = (5t - 1, Cos(t), Sin(t))).
Solution 2:
(i) A parabola.
(ii) An ellipse of eccentricity 2.5 with its major axis parallel to the z axis, its minor
axis parallel to the y axis, and in a plane parallel to the yz-plane with x = 1.
(iii) A circle of radius 5 in the yz-plane centered at the origin.
(iv) A spiral of radius 1 centered on the x-axis.
Problem 3 At what time t does r(t) = (1, t, t2
) intersect x + y + z - 6 = 0?